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A [[Semi-continuous summation method|semi-continuous summation method]] for summing series of numbers and functions, defined by a sequence of functions

\[ g_k(\delta) = \frac{1}{\Gamma(1 + \delta k)}, \quad \delta > 0, \quad k = 0, 1, \dots, \]

where $ \Gamma(x) $ is the gamma-function. A series

\[ \sum_{k=0}^{\infty} u_k \]

is summable by the Mittag-Leffler method to a sum $s$ if

\[ \lim\limits_{\delta \to 0}\sum_{k=0}^{\infty} \frac{u_k}{\Gamma(1 + \delta k)} = s \]

and if the series under the limit sign converges. The method was introduced by G. Mittag-Leffler [[#References|[1]]] primarily for the series

\[ \sum_{k=0}^{\infty} z^k . \]

A Mittag-Leffler summation method is regular (see [[Regular summation methods|Regular summation methods]]) and is used as a tool for the [[Analytic continuation|analytic continuation]] of functions. If $ f(z) $ is the principal branch of an analytic function, regular at zero and represented by a series

\[ \sum_{k=0}^{\infty}a_k z^k \]

for small $z$, then this series is summable by the Mittag-Leffler method to $ f(z) $ in the whole star of the function $ f(z) $ (cf. [[Star of a function element|Star of a function element]]) and, moreover, uniformly in any closed bounded domain contained in the interior of the star.

For summation methods defined by transformations of sequences by semi-continuous matrices $ a_k(\omega) $ of the type

\[ a_k(\omega) = \frac{c_{k+1}\omega^{k+1}}{E(\omega)}, \]

where

\[ E(\omega) = \sum_{k=0}^{\infty} c_k \omega^k \]

is an entire function, Mittag-Leffler considered the case when

\[ E(\omega) = \sum_{k=0}^{\infty} \frac{\omega^k}{\Gamma(1+ak)} \]

A matrix $ a_k(\omega) $ with such an entire function is called a Mittag-Leffler matrix.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Mittag-Leffler,  , ''Atti IV congress. internaz.'' , '''1''' , Rome  (1908)  pp. 67–85</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Mittag-Leffler,   "Sur la répresentation analytique d'une branche uniforme d'une fonction monogène"  ''Acta Math.'' , '''29'''  (1905)  pp. 101–181</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.G. Cooke,   "Infinite matrices and sequence spaces" , Macmillan  (1950)</TD></TR></table>



====Comments====
The function $E(\omega)$ considered by Mittag-Leffler is called a [[Mittag-Leffler function|Mittag-Leffler function]].